11.1 Polynomial Least-Squares Curve Fit

Transcription

1 11.1 Polynomial Least-Squares Curve Fit A. Purpose This subroutine determines a univariate polynomial that fits a given disrete set of data in the sense of minimizing the weighted sum of squares of residuals. The fitted polynomial an be onstrained to math some data points exatly by appropriate setting of the a priori standard deviations of the data errors. Auxiliary subroutines, desribed in Chapter 11.2, may be used to evaluate, differentiate, or integrate a polynomial produed by this fitting proedure. B. Usage B.1 Program Prototype, Single Preision INTEGER M, NMAX, NDEG REAL X( M),Y( M),SD( M or =1),P( NMAX+3), SIGFAC, W( (NMAX+3) (NMAX+3)) LOGICAL SEEKN, COMTRN, CHBBAS Assign values to M, X(), Y(), SD(), NMAX, SEEKN, COMTRN and CHBBAS. If COMTRN =.FALSE. values must also be assigned to P(1) and P(2). CALL SPFIT (M, X, Y, SD, NMAX, SEEKN, COMTRN, CHBBAS, P, NDEG,SIGFAC, W) Following the CALL to SPFIT one may wish to use SCPVAL or SMPVAL to evaluate the fitted polynomial at speifi points, SCPDRV or SMPDRV to differentiate the polynomial, or SCPINT or SMPINT to integrate the polynomial. See Chapter 11.2 for desriptions of these subprograms. B.2 Argument Definitions M [in] Number of data points. X() [in] Array of values of the independent variable. Values need not be ordered and may be repeated. Y() [in] Array of values of the dependent variable indexed to orrespond to the X() values. SD() [in] If SD(1) > 0., eah SD(i) must be positive and will be assumed to be the user s a priori estimate of the standard deviation of the unertainty (observational errors, et.) in the orresponding data value Y(i). If SD(1) < 0., SD(1) will be used as the a priori standard deviation of eah data value Y(i). In this ase the array SD() an be dimensioned SD(1) sine loations following SD(1) will not be used Calif. Inst. of Tehnology, 2015 Math à la Carte, In. NMAX [in] Speifies the maximum degree polynomial to be onsidered. Require NMAX 0. If NMAX > M 1 the subroutine will funtion as though NMAX = M 1. SEEKN [in] If SEEKN =.TRUE. the subroutine will determine the optimum value of NDEG NMAX in the sense desribed below in Setion D. If SEEKN =.FALSE. the subroutine will set NDEG = NMAX unless this is a singular or nearly singular problem, in whih ase NDEG will be redued as neessary. COMTRN [in] If COMTRN =.TRUE. the subroutine will determine the transformation parameters P(1) and P(2) as desribed in Setion D. If COMTRN =.FALSE. the subroutine will use the values of P(1) and P(2) that have been set by the user. CHBBAS [in] If CHBBAS =.TRUE. the subroutine will use the Chebyshev basis. If CHBBAS =.FALSE. the subroutine will use the monomial basis. The Chebyshev basis is reommended for better numerial stability, when NMAX is greater than about six. P() [inout] Parameter vetor defining a polynomial aording to Eqs. (1) and (2) or (3) and (4). Values of P(1) and P(2) may either be set by the user, or by the subroutine, depending upon the value of COMTRN. Values of P(j+3), j = 0,..., NMAX will be omputed by the subroutine. If NDEG < NMAX, the subroutine will set P(i+3) = 0, for i = NDEG + 1,..., NMAX. NDEG [out] Set by the subroutine to indiate the degree of the fitted polynomial. On return NDEG will satisfy 0 NDEG NMAX if the omputation was suessful and will be set to 1 if an error ourred. SIGFAC [out] Number omputed by the subroutine. See disussion of SIGFAC in Setions C and D following. W() [srath] Array of working storage. B.3 Modifiations for Double Preision For double preision usage hange the REAL type statements to DOUBLE PRECISION and hange the subroutine name from SPFIT to DPFIT. Change the names of the auxiliary subroutines mentioned above to DCP- VAL, DMPVAL, DCPDRV, DMPDRV, DCPINT, and DMPINT, respetively. July 11, 2015 Polynomial Least-Squares Curve Fit

2 C. Examples and Remarks C.1 Example Given a set of 12 data pairs (x i, y i ), ompute the weighted least-squares polynomial fit to these data letting the subroutine determine the preferred polynomial degree not to exeed 8. After omputing the leastsquares polynomial p(x), ompute and tabulate the quantities x i, y i, p(x i ), and r i = y i p(x i ). The program DRSPFIT arries out this omputation using subroutines SPFIT and SCPVAL. The output is shown in ODSPFIT. Note that the value of NDEG seleted by the subroutine is 7. C.2 Transformation of the Independent Variable For the best numerial auray, both in determining the best fitting polynomial and in the later evaluation of the polynomial, it is generally advisable to use transformation parameters P(1) and P(2) that ause the transformed variable u of Eqs. (1) or (3) to range from 1 to +1. This ondition an be obtained by setting COMTRN =.TRUE. When fitting with polynomials of degree higher than about six, the Chebyshev basis generally gives better numerial auray than the monomial basis. To obtain the potential numerial advantages of the Chebyshev basis it is essential to ause the transformed variable, u, to range from 1 to +1. To fore an identity transformation (i.e., effetively no transformation) of the independent variable, set COMTRN =.FALSE., P(1) = 0., and P(2) = 1. C.3 Interpretation of SIGFAC If the user has set all SD(i) = 1.0, or equivalently, set SD(1) = 1.0, SIGFAC an be interpreted as an a posteriori estimate of the standard deviation of the error in eah Y(i) value. More generally, whatever values the user has assigned to the SD(i) s, the a posteriori estimate of the standard deviation of the error in Y(i) is SIGFAC SD(i). C.4 Equality Constraints If the user wishes the fitted polynomial to agree to mahine auray with one or more of the data points, this an be aomplished by setting the SD() value for suh points muh smaller than for the other points. Let ɛ denote the mahine preision, i.e., ɛ = R1MACH(4) in single preision or ɛ = D1MACH(4) in double preision, see Chapter For some value of in the range , we suggest setting SD(i) = ɛ for the points at whih an exat fit is desired and SD(i) = 1.0 for all other points. For example if ɛ = 10 8 set SD(i) in the range 10 4 through 10 6 for the exat fit points. Using < 0.5 may not produe suffiiently small residuals at the desired points of exat fit, whereas setting > 0.75 may trip the near-singularity test in these subroutines leading to an unwanted alteration of the problem. Note that it is not mathematially reasonable to attempt to fore exat fits at more than NMAX+1 points. D. Funtional Desription The user provides data (x i, y i, s i ), for i = 1,..., M, and an integer NMAX. If SEEKN =.FALSE. and the problem is not rank-defiient or extremely ill-onditioned, the subroutine simply determines the polynomial p n (x) of degree n = NMAX that minimizes the following weighted sum of squares of residuals: M [ ] 2 ρ 2 yi p n (x i ) n = i=1 s i The subroutine also omputes [ ρ 2 n σ n = max(1, M n 1) ] 1/2 and returns this value as SIGFAC. The subroutine makes use of subroutines SROTMG and SROTM (or DROTMG and DROTM) from the BLAS1 (See Chapter 6.3.) to implement a sequential Modified Givens orthogonal transformation method of reduing the matrix of the least-squares problem to triangular form [1], and then solves the triangular system. This method has exellent numerial stability. After triangularizing the matrix of the problem for degree NMAX, the quantities ρ 2 n and σ 2 n may be omputed inexpensively for all degrees n from zero through NMAX. Thus, if the user sets SEEKN =.TRUE. the subroutine omputes these values of ρ 2 n and omputes CMIN = min{σ 2 n : 0 n NMAX}. The quantity ρ n is a stritly noninreasing funtion of n whereas σ n typially dereases initially with inreasing n but then osillates and slowly inreases as n beomes so large that the polynomial p n (x) is fitting noise rather than a true polynomial signal in the data. The subroutine sets NDEG to be the smallest value of n for whih σn CMIN, and it sets SIGFAC = σ NDEG. This method of seleting NDEG is generally satisfatory when the ratio NMAX/M is reasonably small, say less than 1/3. If this ratio is larger there is a tendeny for the NDEG seleted by this method to be larger than one might pik if one viewed graphs of solution polynomials of different degrees Polynomial Least-Squares Curve Fit July 11, 2015

3 Whatever the setting of SEEKN, if the problem for degree NMAX is rank-defiient, or very ill-onditioned, the subroutine will restrit onsideration to lower degrees for whih the problem is not extremely ill-onditioned. In partiular this redution of degree will ertainly our if NMAX+1 > M. Parameterization of the polynomial, p n (x) In this subroutine an n th degree polynomial p n (x) is represented in one of two speial parametri forms (monomial basis or Chebyshev basis), both of whih inlude the speifiation of a linear transformation of the independent variable in the parameterization. The parameters defining an n th degree polynomial p n (x) are stored in the array P(i), i = 1,..., n + 3. Using the monomial basis these parameters define the polynomial p n (x) as follows: u = x P (1) (1) P (2) n p n (x) = P (i + 3)u i (2) i=0 whereas, if the Chebyshev basis is requested, p n (x) is defined as: u = x P (1) (3) P (2) n p n (x) = P (i + 3)T i (u) (4) i=0 where T i (u) denotes the Chebyshev polynomial of degree i. The Chebyshev polynomials may be defined as follows: T 0 (u) = 1, T 1 (u) = u, T 2 (u) = 2u 2 1 T i (u) = 2u T i 1 (u) T i 2 (u) i = 3, 4,... The parameters P(1) and P(2) that our in the transformation formula (1) or (3) may be set by the user (COMTRN =.FALSE.) or else omputed by this subroutine (COMTRN =.TRUE.). In the latter ase P(1) and P(2) will be omputed as P(1) = (x max + x min )/2 P(2) = (x max x min )/2 where x max and x min are respetively the maximum and minimum values of x in the data array X(i), i = 1,..., M. This auses the transformed variable u of Eq. (1) or (3) to range from 1 to +1 as x ranges from x min to x max. Referenes 1. Charles L. Lawson and Rihard J. Hanson, Solving Least-Squares Problems, Prentie-Hall, Englewood Cliffs, N. J. (1974) 340 pages. E. Error Proedures and Restritions The use of the Chebyshev basis is effetive in improving the onditioning of the problem only if a transformation of the variable is used that maps the interval of interest for both fitting and evaluation to an interval approximately oinident with [ 1., 1.]. The automati proedure for seleting the degree of the fitted polynomial, used when SEEKN =.TRUE., tends to selet the degree somewhat higher in some ases than one would hoose by looking at plots of fits of different degrees. This subroutine will issue an error message using the error proessor in Chapter 19.2 at level 0, set NDEG = 1, and return if any of the following erroneous onditions exists: 1. SD(1) = 0, or M 0, or NMAX < COMTRN =.FALSE. and P(2) = SD(1) > 0. and some SD(i) 0. for 1 < i M. F. Supporting Information The soure language is Fortran 77. Entry Required Files DPFIT AMACH, DERM1, DERV1, DPFIT, DROTM, DROTMG, ERFIN, ERMSG, IERM1, IERV1 SPFIT AMACH, ERFIN, ERMSG, IERM1, IERV1, SERM1, SERV1, SPFIT, SROTM, SROTMG Initially designed and programmed by C. L. Lawson, JPL, Modified by Lawson and S. Y. Chiu, 1984, to use Modified Givens rather than Householder transformations to redue the storage requirement for W() and the omplexity of the program. Also, adapted the ode to Fortran 77. July 11, 2015 Polynomial Least-Squares Curve Fit

4 DRSPFIT program DRSPFIT >> DRSPFIT Krogh S p e i a l ode f o r C onversion. >> DRSPFIT Krogh Added e x t e r n a l statement. >> DRSPFIT Krogh Delare a l l v a r i a b l e s. >> DRSPFIT Krogh Remove 0 i n format ( ag ai n?) >> DRSPFIT Krogh Changes t o use M77CON >> DRSPFIT WVS Remove 0 i n format >> DRSPFIT CLL E d i t e d f o r Fortran 90 >> DRSPFIT Lawson I n i t i a l Code. S r e p l a e s? : DR?PFIT,?PFIT,?CPVAL Demonstration d r i v e r f o r SPFIT. ++ Code f o r.c. i s i n a t i v e %% l o n g i n t j ; ++ End external SCPVAL real X( 1 2 ),Y( 1 2 ),P( 1 1 ),SIGFAC,W( ),SCPVAL real SIG ( 1 ), R, YFIT integer I, M, NDEG, NP3 data X / 2. E0, 4. E0, 6. E0, 8. E0, 1 0. E0, 1 2. E0, 1 4. E0, 1 6. E0, 1 8. E0, 2 0. E0, 2 2. E0, 2 4. E0 / data Y / 2. 2 E0, 4. 0 E0, 5. 0 E0, 4. 6 E0, 2. 8 E0, 2. 7 E0, 3. 8 E0, 5. 1 E0, 6. 1 E0, 6. 3 E0, 5. 0 E0, 2. 0 E0 / data SIG ( 1 ) / 1.E0 / data M / 12 / 1001 format ( 1X/ I X Y YFIT R=Y YFIT /1X) 1002 format (1X, I2, F6. 0, 2 F9. 3, F10. 3 ) a l l SPFIT(M,X,Y, SIG, 8,.TRUE.,.TRUE.,.TRUE.,P,NDEG, SIGFAC,W) NP3 = NDEG+3 ++ Code f o r.c. i s a t i v e print (/ NDEG =, I2, 1 0X, SIGFAC =, F8. 4 / / P( 1 ),P( 2 ) =,9X, 2 F15. 8 / / P ( 3 ),..., P(NDEG+3) =,3 F15. 8 / (21X, 3 F15. 8 ) ), NDEG, SIGFAC, ( P( I ), I =1,NP3) ++ Code f o r.c. i s i n a t i v e %% p r i n t f ( %% \n NDEG =%2l d SIGFAC =%8.4 f, ndeg, s i g f a ) ; %% p r i n t f ( %% \n\n P( 1 ),P(2) = %15.8 f %15.5 f \n\n P ( 3 ),..., P(NDEG+3) =, %% p [ 0 ], p [ 1 ] ) ; %% f o r ( i = 2 ; i < ( np3 ) ; i +=3){ %% f o r ( j = i ; j < ( i < np3 2? i+3 : np3 ) ; j++) %% p r i n t f ( %15.8 f, p [ j ] ) ; %% i f ( i < np3 2) p r i n t f ( \n ) ; } %% p r i n t f ( \n ) ; ++ End write (, ) do 20 I = 1,M YFIT = SCPVAL(P,NDEG,X( I ) ) R = Y( I ) YFIT write (, ) I,X( I ),Y( I ),YFIT,R 20 ontinue stop end Polynomial Least-Squares Curve Fit July 11, 2015

5 ODSPFIT NDEG = 7 SIGFAC = P( 1 ),P( 2 ) = P ( 3 ),..., P(NDEG+3) = I X Y YFIT R=Y YFIT July 11, 2015 Polynomial Least-Squares Curve Fit